This low-key but frequently hilarious comedy begins with Simon Amstell awkwardly telling his family that he wants to quit as the acerbic host of Never Mind the Buzzcocks. His family, assembled at his grandmother's house with the pop panel show on in the background, are shocked. "Do you want a banana?" his gran asks.

The show arrived on BBC2 in 2010, when Amstell was doing what he's best known for – "taking the piss out of pop stars", as his screen mother Tanya puts it. Amstell's venture into acting and writing was a brave move, especially when the show in question was a semi-autobiographical sitcom. Unfolding in a chintzy Essex living room, the first episode tackles his inexperience right away. "People congratulate me for being mean, it's not very Buddhist," says Amstell who, away from the scripted gags, seems shy and more than a little pretentious. "I just want to do something meaningful. Maybe I'll write something or act a bit." The incredulous response – "Act?" – from his mother, brilliantly played by Rebecca Front, is utterly damning. And her misgivings were echoed by critics at the time.

Yet, while Daniel Day-Lewis probably won't be losing any sleep over his future Oscar chances, Amstell does improve across the six episodes that make up series one of Grandma's House, eventually reaching a point where he convincingly pulls off the role of Simon Amstell, former host of Never Mind the Buzzcocks. Things unfold slowly in the suburban home of Amstell's Jewish grandparents, whom he visits weekly alongside his mother, aunt and younger cousin. The programme sweetly captures the regression so many adults feel when stepping back into childhood homes – as well as the incessant bickering, most of which occurs over Amstell's dislike of Clive, his mum's new boyfriend.

Fussed over by his family, he will always be – to them anyway – the talented and precocious child who just wanted to be famous. They just can't get their heads round his change of heart, his struggle to reconcile fame with basically being horrible to people for a living, and the general sense that he should be doing something more significant. As his mum puts it: "You're on TV. What more do you want?"

That's a question Amstell never really answers. Fans of his standup will recognise the character more easily than those who have binged on old Buzzcocks episodes on Dave. Gentle and tied up with neuroses, he is a man in love with the idea of self-improvement and fulfilment, with little notion of what they actually mean or how to get them. His family smothers him in love – and doesn't understand a thing he says. When his grandad thinks he may have cancer, Amstell advises him to beat the illness by thinking positively. Nobody tells him he's wrong, but nobody listens either. He ends up like a plaything for the family. He wants to write a play and read self-help books. They want to hear that impression of Mr Blobby he used to be so good at.

Co-written by Dan Swimer, Grandma's House echoes The Royle Family in places, with its sense of family and sentimentality, as well as the simple fact that most scenes are played out from the comfort of a sofa in front of the TV (watching an old episode of Buzzcocks, naturally). The show ran for one further series but last year Amstell confirmed there were no plans for a third. "We felt we'd written all the pain out of us," he said.